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> Norman Samish wrote:
>>
>> Thanks to all who replied to my question. This question has
>> bothered me for years, and I have hopes that some progress can
>> be made towards an answer.
>>
>> I've heard some interesting concepts, including:
>> (1) "Numbers must exist, therefore 'something' must exist."
>> (2) "Something exists because Nothingness cannot non-Exist."
>>
>> Perhaps the above two are equivalent.
>>
>> With respect to (1) above, why must numbers exist?
>
> I am not sure that any definitive answer can be given to this
> question. A possible argument is that the existence of numbers
> by themselves is much easier to accept than the existence of
> "usual material things" in a classical sense. Of course, even
> if it was really a weakest assumption, it is not granted.
What can be said about numbers is that it is impossible to explain what
numbers are to someone who does not already knows what they are.
If a TOE does not implicitly or explicitly presupposes the existetnce
of natural numbers, then the natural numbers will not be definable in
that TOE, and for this reason that TOE will not be a plausible TOE,
although Hartree Field, if I remember correctly, makes a case for a
science without number. (Of course such a science will be aristotelian
and un-platonist automatically).
That is why I put "arithmetical realism" explicitly in the definition
of comp ("yes doctor" + "Church thesis + Arithmetical realism).
Once you accept the existence of natural numbers, and of addition,
multiplication; you get freely universal dovetailing, and thus you get
enough for comp-like kind of TOE.
For the notion of observer or lobian entities, you need the belief in
numbers, addition and multiplication; but also of some amount of
induction, that is, the axioms saying, for any formula A(x) already
definable, that :
{A(0) and for all n [A(n) -> A(n+1)]} implies that for all n A(n).
(In english, the induction axioms say that if some property is true for
the number 0, and remains true when going from any number n to the
successor of n, then the property will be true for all number n. Think
about an infinite range of dominoes).
Those induction axioms are enough to explain why lobian machine will
have enough introspection ability to discover their ignorance and to
reflect about.
So numbers are needed. Are numbers enough? Ontologically: perhaps (and
even surely with some Occam razor in case comp is accepted).
Epistemologically: No. By incompleteness a lot of apparently simple
problems needs theories far more powerful than Peano Arithmetic (say)
to find solution. The use of analytical tools (complex analysis) in
number theory is a symptom of that phenomenon.
Just to explain the behavior of natural numbers you need the whole
mathematics.
> The idea behind "numbers must exist" is that "God Himself
> cannot make that two plus two equates something different of
> four". Another way to say it is that "even if there were nothing
> (or no thing) there would remain that whenever/wherever there
> would be something in which natural numbers could be thought of,
> the Fermat conjecture should be true". If natural numbers did
> not exist, this necessity would immediately apply to them
> whenever and wherever they appear. I would say that the set of
> such necessities is not different from natural numbers themselves.
I agree.
> Of course, it is too strong to claim that natural numbers exist
> individually and one independently of another. The arguement is
> that "arithmetics" as a whole exists by itself (and as something).
Yes. It would be ridiculous and actually inconsistent to believe that
the number 13 can be suppressed.
> Yes. It should even forbid the existence of a "Fermat theorem
> constraint" and it is hard to imagine (at leat for me) that such
> a constraint could not exist. So it is not so puzzling (at least
> to me) that something exist.
Mmmhhh... By accepting the numbers I would agree. Nevertheless, our
very ability to "understand" numbers is, for me, as mysterious as the
understanding of a word like "consciousness" itself.
I feel a little bit like Colin McGuin and tend to feel that numbers are
sort of a necessary mystery.
Bruno
http://iridia.ulb.ac.be/~marchal/
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